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Figure Q2 shows a block diagram with a plant Gp(s) which is subjected to a disturbance R(s) and the reference input R(s). The response C(s) of the plant is governed by a controller G.(s). For the situation where there is no reference input (i.e. R(s) = 0), derive the closed loop transfer function relating the response C(s) to the disturbance D(s). Similarly derive the transfer function relating the response to the reference input R(s) and hence the response of the system to both inputs. (a) (b) The plant may be described by the transfer function 1 Gp(s) = · s(s + 7) It is proposed to apply a proportional plus derivative controller to the system such that G.(s) = Kp + Kas For the case where D(s) = 0, calculate the values of the gains K, and Ka if the performance requirements of the system are that it should have maximum overshoot of 15% and a natural frequency of 9 rad/s. Integral action with gain K; is added to the controller to form a PID controller. Find the values of K; for which the system remains stable. (c)

Figure Q2 shows a block diagram with a plant Gp(s) which is subjected to a disturbance R(s) and the reference input R(s). The response C(s) of the plant is governed by a controller G.(s). For the situation where there is no reference input (i.e. R(s) = 0), derive the closed loop transfer function relating the response C(s) to the disturbance D(s). Similarly derive the transfer function relating the response to the reference input R(s) and hence the response of the system to both inputs. (a) (b) The plant may be described by the transfer function 1 Gp(s) = · s(s + 7) It is proposed to apply a proportional plus derivative controller to the system such that G.(s) = Kp + Kas For the case where D(s) = 0, calculate the values of the gains K, and Ka if the performance requirements of the system are that it should have maximum overshoot of 15% and a natural frequency of 9 rad/s. Integral action with gain K; is added to the controller to form a PID controller. Find the values of K; for which the system remains stable. (c)

Introductory Circuit Analysis (13th Edition)
Introductory Circuit Analysis (13th Edition)
13th Edition
ISBN:9780133923605
Author:Robert L. Boylestad
Publisher:Robert L. Boylestad
Chapter1: Introduction
Section: Chapter Questions
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Figure Q2 shows a block diagram with a plant G,(s) which is subjected to a disturbance R(s) and the reference input R(s). The response C(s) of the plant is governed by a controller G.(s). For the situation where there is no reference input (i.e. R(s) = 0), derive the closed loop transfer function relating the response C(s) to the disturbance D(s). Similarly derive the transfer function relating the response to the reference input R(s) and hence the response of the system to both inputs. (а) (b) The plant may be described by the transfer function 1 Gp (s) = s(s + 7) It is proposed to apply a proportional plus derivative controller to the system such that G.(s) = Kp + Kas For the case where D(s) = 0, calculate the values of the gains K, and K if the performance requirements of the system are that it should have maximum overshoot of 15% and a natural frequency of 9 rad/s. (c) Integral action with gain K; is added to the controller to form a PID controller. Find the values of K; for which the system remains stable. (d) When K; = 225, the system has a real pole at s = -3.74 and a pair of complex poles such that the characteristic equation may be described by (s + 3.74)(s² + 23wns + wh) = 0 Calculate the damping ratio 3 and natural frequency wn of these complex poles. D(s) R(s) C(s) Ge(s) Gp(8) Figure Q2.
Transcribed Image Text:Figure Q2 shows a block diagram with a plant G,(s) which is subjected to a disturbance R(s) and the reference input R(s). The response C(s) of the plant is governed by a controller G.(s). For the situation where there is no reference input (i.e. R(s) = 0), derive the closed loop transfer function relating the response C(s) to the disturbance D(s). Similarly derive the transfer function relating the response to the reference input R(s) and hence the response of the system to both inputs. (а) (b) The plant may be described by the transfer function 1 Gp (s) = s(s + 7) It is proposed to apply a proportional plus derivative controller to the system such that G.(s) = Kp + Kas For the case where D(s) = 0, calculate the values of the gains K, and K if the performance requirements of the system are that it should have maximum overshoot of 15% and a natural frequency of 9 rad/s. (c) Integral action with gain K; is added to the controller to form a PID controller. Find the values of K; for which the system remains stable. (d) When K; = 225, the system has a real pole at s = -3.74 and a pair of complex poles such that the characteristic equation may be described by (s + 3.74)(s² + 23wns + wh) = 0 Calculate the damping ratio 3 and natural frequency wn of these complex poles. D(s) R(s) C(s) Ge(s) Gp(8) Figure Q2.
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